We estimate the singular set of solutions to elliptic systems with linear growth of the type $$-div a(x,u,Du) = b(x,u,Du)\;,$$ where $a(x,u,Du)$ depends in a Dini continuous way on $(x,u)$; this is the natural limit case of Hölder continuity of the coefficients for proving partial continuity of the gradient, as shown by Duzaar & Gastel (Archiv der Mathematik, 2002). The singular set is estimated using Hausdorff measures generated by functions which are not powers.